Memory and memoryless systems; causality

Systems with memory and memoryless systems

A system is memoryless (static) if its output at any time $n$ depends only on the input at the same time $n$. A system with memory (dynamic) has an output that depends on past and/or future input values (or past outputs), so it must store information.

For an LTI system, $y[n]=\sum_k h[k]x[n-k]$. This depends only on $x[n]$ if $h[k]=0$ for all $k\ne0$. So a DT LTI system is memoryless only if

$$h[n]=K\delta[n] \quad\Rightarrow\quad y[n]=Kx[n]$$

Any other $h[n]$ (e.g. $h[n]=\delta[n]-\delta[n-1]$) gives a system with memory.

Causality of discrete-time LTI systems

A system is causal if its output at time $n$ depends only on the present and past inputs ($x[n], x[n-1],\dots$), not on future inputs. Causal systems are "non-anticipative"; all real-time physical systems are causal.

For an LTI system,

$$y[n]=\sum_{k=-\infty}^{\infty}h[k]\,x[n-k]$$

The terms with $k<0$ use $x[n-k]$ with $n-k>n$, i.e. future inputs. For the output not to depend on them, those terms must vanish for every input:

$$\boxed{h[n]=0 \quad \text{for } n<0}$$

This is the necessary and sufficient condition. The convolution sum of a causal LTI system then becomes

$$y[n]=\sum_{k=0}^{\infty}h[k]\,x[n-k]=\sum_{k=-\infty}^{n}x[k]\,h[n-k]$$

Also, for a causal LTI system, initial rest holds: if $x[n]=0$ for $n<n_0$, then $y[n]=0$ for $n<n_0$.

Examples

• $h[n]=(0.5)^n u[n]$: zero for $n<0$, so causal.
• $h[n]=\delta[n+1]+\delta[n]$, i.e. $y[n]=x[n+1]+x[n]$: $h[-1]=1\ne0$, so non-causal.
• Accumulator $h[n]=u[n]$: causal, with memory.
• Ideal low pass filter $h[n]=\frac{\sin\omega_c n}{\pi n}$: non-zero for $n<0$, so non-causal (cannot work in real time).

Note: a memoryless system is always causal, but a causal system need not be memoryless.